1 EXISTENCE AND UNIQUENESS OF HAAR MEASURE 243
(14.1.1) (Y(4/X*) =f(s-*x)9 (8(5)/)(jc) =/(jcs).
It follows immediately from this definition that we have
(14.1 .1 .1 ) y(*0/= YO)(Y(0/), S(*0/
for all j, t e G.
If /j is a (complex) measure on G, we denote by y(s)n and $C?)ju the meas-
ures on G which are the images of fi under the homeomorphisms x\-^sx and
1, respectively (13.1.6), so that
for all functions /e Jfc(G). From this definition it follows that
(14.1 .2.1 )
for all s, t e G.
The measure /i is said to be left (resp. right) invariant if
(1 4.1 .2.2) y(s)ju = /* (resp. 8(s)n = /x)
for all s e G.
If a measure jj. ^ 0 on G is left invariant, then Supp(/x) = G, because
Supp(j(s)fj) = ^ • Supp(|i) for all s e G by virtue of (1 3.1 9.4), and Supp(ju) ^ 0.
Similarly for right-invariant measures.
Let ^ be a left-invariant measure on G and let/be a //-integrable mapping
of G into R or C. Then for each ,seG the function xt-+f(s~lx) is also
/i-integrable, and we have
by (13.7.10). In particular, if A is a ju-integrable set, then so is ^A, and
(22.214.171.124) /x(jA) = ^(A).
If /is any mapping of G into a set E, we write
(14.1.3) /(*) ^/(jt-1) for all xeG. EXISTENCE AND UNIQUENESS OF HAAR MEASURE